Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 136 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 


Ë} ÒÒ}ÈÁË ¯©m©° Ë®äÈËäÈÒ}Òlnj
ÙkÓÈãÒÒË°}È«˺äË¯Ò«ÒãÒÓË®ÓÈ«ÈãË¯Èµ

äÓºmkp
vãË°mÒË

cÈÏãºÎËÓÒË º¹¯ËËãÒËã« ¹º
i
 °ºã ÒäËË mÒ
∑
=
+
−=
n
k
i
k
ki
ik
MA
1
)1(det
α
ÒãÒ
i
k
n
k
i
k
ik
MMA
∑
=
+
−=
1
)1(det
˺¯ËäÈ

jäËË äË°º ¯ÈmËÓ°mº
∆⋅=
∑
=
js
n
i
isij
D
δα
1
 Ë
∆=
det A
Ò
≠
=
=
sj
sj
js
,0
,1
δ
iº}ÈÏÈËã°mº
ºº¹¯ËËãËÓÒÈãË¯ÈÒË°}ººº¹ºãÓËÓÒ«ÒäËËä
njnjjjjj
DDDA
ααα
+++=
...det
2211

ºË°m˯ÎËÓÒË˺¯Ëä©ã«°ãÈ«
j=s
°¹¯ÈmËãÒmº
° ˹˯
j≠s
 ºÈ m©¯ÈÎËÓÒË
nsnjsjsj
DDD
ααα
+++ ...
2211
äºÎÓº ¯È°
°äÈ¯ÒmÈ}È}¯ÈÏãºÎËÓÒË¹º
s
ä°ºã º¹¯ËËãÒËã«äÈ¯Ò©}ºº¯º®
s
®°ºãË°ºm¹ÈÈË °
j
ä°ºãºä sº È}º®º¹¯ËËãÒËã¯ÈmËÓÓã¹º
°ãË°mÒ
˺¯ËäÈº}ÈÏÈÓÈ
vãË°mÒË

p°ãÒ äÈ¯ÒÈ
A
ÓËm©¯ºÎËÓÈ º ªãËäËÓÈäÒ º¯ÈÓº® äÈ¯Ò©
A
−1
«mã«°«Ò°ãÈ
],1[,;
)1(
nji
M
i
j
ji
ij
=
∆
−
=
+
β

iº}ÈÏÈËã°mº
sÈ®Ëä ¹¯ºÒÏmËËÓÒË äÈ¯Ò
A
Ò
B
 Ë
B
ÒäËË ªãËäËÓ©
],1[,;
nji
ij
=
β
 È
°Èäº ¹¯ºÒÏmËËÓÒË  ªãËäËÓ©
pq
γ
 ºÈ °ºãÈ°Óº
º¹¯ËËãËÓÒÒ˺¯ËäË
],1[,;
11
)1(
111
njiD
M
pqpq
n
j
qjpj
n
j
j
q
qj
pj
n
j
jqpjpq
==⋅∆⋅
∆
=
∆
=
∆
−
==
∑∑∑
==
+
=
δδααβα
γ

kÓÈ㺠ÒÓºË°ººÓºËÓÒË¹ºãÈË°«Òã«¹¯ºÒÏmËËÓÒ«
A B

 Ë }  Ò Ò    } È Á Ë  ¯ ©   m © °  Ë ®   ä È ˆ Ë ä È ˆ Ò } Ò   l n ‘ j 
ÙkÓÈã҈ÒË°}È«˺äˈ¯Ò«ÒãÒÓË®ÓÈ«ÈãË­¯ÈµäÓºmkp



  vãË°ˆmÒË        cÈÏãºÎËÓÒË º¹¯ËËã҈Ëã« ¹º i‚ °ˆºã­‚ ÒäËˈ mÒ
  
                                 n                                                     n
                         det A = ∑ (−1) k + i α ki M k ÒãÒ det A = ∑ (−1) k + i M ki M k 
                                                               i                                                     i

                                         k =1                                               k =1
              
              
              
                                                                            n
 ‘˺¯ËäÈ
 
                         jäËˈ          äË°ˆº       ¯ÈmËÓ°ˆmº            ∑ α ij Dis = δ js ⋅ ∆          Ë        ∆ = det A  Ò
                                                                            i =1
                                1, j = s
                         δ js =          
                                0, j ≠ s
        
  iº}ÈÏȈËã°ˆmº
   
   
      ºº¹¯ËËãËÓÒ ÈãË­¯ÈÒË°}ººº¹ºãÓËÓÒ«ÒäËËä
          

                                                  det A = α1 j D1 j + α 2 j D2 j + ... + α nj Dnj 
                                                                             

          ˆºË°ˆ ‚ˆm˯ÎËÓÒˈ˺¯Ëä©ã«°ã‚È«j=s°¹¯ÈmËãÒmº
          
          ‚°ˆ  ˆË¹Ë¯  j≠s  ‘ºÈ m©¯ÈÎËÓÒË α1 j D1s + α 2 j D2 s + ... + α nj Dns  äºÎÓº ¯È°
          °äȈ¯ÒmȈ }È}¯ÈÏãºÎËÓÒ˹º sä‚°ˆºã­‚º¹¯ËËã҈Ëã«äȈ¯Ò©‚}ºˆº¯º®
          s® °ˆºã­Ë °ºm¹ÈÈˈ ° jä °ˆºã­ºä sº ˆÈ}º® º¹¯ËËã҈Ëã  ¯ÈmËÓ ӂã  ¹º
          °ãË°ˆmÒ 
     
     
     ‘˺¯ËäȺ}ÈÏÈÓÈ
              
              
              
 vãË°ˆmÒË              p°ãÒ äȈ¯ÒÈ A  ÓËm©¯ºÎËÓÈ ˆº ªãËäËӈÈäÒ º­¯ÈˆÓº® äȈ¯Ò©
 
                                                              (−1) i + j M j
                                                                                       i
                                 −1
                             A        «mã« ˆ°«Ò°ãÈ β ij =                ; i, j = [1, n] 
                                                                     ∆
              
    iº}ÈÏȈËã°ˆmº
     
          sÈ®Ëä ¹¯ºÒÏmËËÓÒË äȈ¯Ò                                 A  Ò        B  Ë            B  ÒäËˈ ªãËäËӈ©
              β ij ; i, j = [1, n]  È °Èäº ¹¯ºÒÏmËËÓÒË  ªãËäËӈ© γ pq  ‘ºÈ °ºãÈ°Óº
          º¹¯ËËãËÓÒ ÒˆËº¯ËäË
          
                                                     (−1) j + q M q 1 n
                         n                  n                       j
                                                                                1
                  γ pq = ∑α pj β jq = ∑α pj                        = ∑α pj Dqj = ⋅ ∆ ⋅ δ pq = δ pq ; i, j = [1, n] 
                        j =1               j =1            ∆        ∆ j =1      ∆
                                                                             
          kÓÈ㺠ÒӺ˰ººˆÓº ËÓÒ˹ºã‚Èˈ°«Ò㫹¯ºÒÏmËËÓÒ« A                                                 B 
          

Страницы